Benjamin, their first baby after Lucas, made it to 24 weeks and lived for 16 days in Dunedin Hospital in October 2012.
The next baby, Finn, was stillborn at 19 weeks.
The next, Philip, was stillborn at 20 weeks despite the first attempt to sew up the cervix.
"It's been terrible," Mrs White said.
The ordeal has been made harder by the fact that they have no close relatives in New Zealand. Mrs White, 36, is from Germany and married in 2010 after meeting Mr White on holiday in Tasmania.
Mr White, 54, a deerfarmer near Winton who had never married before, is an only child and his parents are dead. The couple want to try everything they can to give Lucas a brother or sister.
"I think if I'm sitting here in 10 years, I want to know what if?" Mrs White said. "It's now definitely our last chance."
Their specialist has told them their only options now are surrogacy or adoption. But there is a shortage of babies for adoption and they believe they would not be a high priority because they already have one child, so they googled surrogacy and found that other people had used Facebook to find a surrogate mother.
"I had never been on Facebook," Mrs White said. "So I joined, and I asked the people from Lucas' play group if they could help me to spread it around because I didn't have any Facebook friends."
The play group parents have done a good job. The Facebook page, "Southland couple looking for surrogate in New Zealand", went up in the middle of last week and drew 366 likes by late yesterday.
About "half a dozen" women have already offered to be surrogate mothers. The couple has had to turn some away because they are too young, and an ethics committee guidelines for approving a surrogacy agreement is that the surrogate mother should have "completed her family". But they are still talking to "two or three".
Surrogacy rules
• Surrogate mothers in NZ cannot be paid, except for counselling and legal costs.
• Both surrogate mother and intending parents must get independent legal and medical advice and counselling.
• The surrogate mother is the legal mother of the child and can only transfer rights to intending parents if they legally adopt the child.
• Any advance agreement to adopt the child is not legally enforceable, so the surrogate mother has the legal right to keep the child if she chooses.
• Surrogacy must be approved by the Ethics Committee on Assisted Reproductive Technology.
• The committee tries to minimise the chances that the surrogate will want to keep the child through a guideline that the surrogate should have "completed her family".
• The committee received 12 surrogacy applications in the year to June 2014.